416 STRENGTH OF MATERIALS which set becomes proportional, or nearly so, to the distortion produced by the applied force. It is generally about one half or one third the ultimate strength. 5. Tensile resistance, or tenacity, is determined by experiment for each material. The ultimate strength or breaking load of any piece is measured by the product of the area of fractured section into the te- nacity of the material of which it is composed ; i. e., P=TK, and K=-J where P represents the breaking force, T the tenacity of the ma- terial, and K the area of section. Values of T are given in the accompanying table of co- efficients of resistance. The very best grades should have values 20 per cent, higher. P and T are taken in pounds upon the square inch. 6. When thin cylinders are exposed to internal pressure, as in steam boilers, steam cylinders, &c., the bursting pressure may be determined by multiplying the thickness of the shell by the tenacity of the material, as given above, and dividing by the semi-diame- ter. To ascertain the thickness, the pressure and the diameter of the cylinder being given, multiply the pressure by the semi-diameter, and divide by the tenacity of the material as given in the table; or P= *, and t=~, where P = pressure, t = thickness, T = tenacity, and r = radius of the cylinder. If d = diameter, P=2^, and t=^. Where the joints are double-riveted, the strength at the joints is usually about O'Y that of the solid plate ; sin- gle-riveted joints have 0'56 the strength of a solid plate. The mean strength of single-riv- eted boilers, where the joints are properly shifted, or " broken," is nearly or quite five eighths that of solid plate. A sphere will bear twice as much internal pressure as a cylin- der of the same diameter. A thick cylinder is not as strong relatively as a thin cylinder of the same material, and no cylinder, how- ever thick, can withstand an internal pressure exceeding its limit of tenacity, T, as given above. The rule for calculating the strength of a thick cylinder is expressed algebraically thus : J >:= Tj^-j-^, and = ^(T^P)' w ^ ere r = i n " ternal radius, R = the external radius, and other values as before. For thick spheres, 7. The stayed surfaces are usually the strongest parts of a steam boiler. The following formula gives the proper distance between stays, where t thickness of plate, P = the pres- sure in pounds per square inch, and F= the factor of safety: d=^. P=F/^Y. 8. 5FP d / Bolts, rivets, and lugs are usually exposed to shearing strain. The resistances to shearing of the most commonly used materials are to be taken as equal to the tensile strength. p_ T 2R-2r r - Where shearing is to be resisted, the parts should be fitted with great care, to avoid the possibility of cutting, and to insure that all parts of the cross section attacked shall resist the shearing force as nearly as possible to- gether. Where a pin is fitted but not forced into its socket, the resistance to shearing is taken as three fourths of that due the sec- tion exposed to rupture. 9. Crushing is re- sisted by any given material with a force that varies very greatly with the form given it. Very short columns or compact masses resist very high crushing strains, in consequence of the resistance offered by their particles to dis- persion, as well as by their cohesion. Tall columns first bend and then break under a comparatively slight force. The figures in col- umn C of the table give the resistance to crush- ing when bending does not occur. Seasoned timber has nearly twice as great resistance to crushing as green. Steel should not be used under pressure exceeding its compressive elas- ticity, which, in tool steel, is about 50,000 Ibs. to the square inch. Wrought iron should not be used under pressure exceeding 25,000 Ibs. to the square inch. 10. For tall columns, the following formulas were proposed by Prof. Eaton Hodgkinson : MATERIALS. Rounded ends. Flat ends. Solid cylindrical cast-iron col- umns Hollow cylindri- cal cast-iron columns Solid cylindrical wrought- iron columns Solid square pil- lar of Dantzic W=13 ~f J)3-55 L 2 T)4 W-10-9 oak L a Solid square pil- | lar of red > w _ 7g D* pine ) L a In these formulas W = crushing weight in tons, D = outside diameter in inches, d = inside diameter in inches, L = length in feet, and D <'4L; i. e., the columns are more than 30 diameters in length. Prof. Rankine gives for tall columns. P = crushing load in pounds ; S = sectional area in square inches ; I = length and D = external diame- ter, both being in the same units of measure, whether feet or inches. The following are the values of /"and a: MATERIALS. Value of/. Value of a. Forms of column. Oast iron 80.000 86,000 86.000 86,000 86,000 86,000 36,000 50,000 6,500 i*. Sooo
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525o mn tH Hollow cylinder. Solid rectangle. Thin square tube. Solid cylinder. Thin cylinder. Ande iron. + shaped. For all shapes. Solid rectangle. Wrought iron. . Best American wrought iron . . . Timber
